AP225

About me

Fun facts on soft matter

After being bombarded with particle physics and condensed matter for 4 and a half years of undergraduate education (ah! the greek educational system!), honestly all of soft matter is fun to me simply because it is so so tangible!

Final Project

The project I chose is only a part of the 'bigger picture' that interests me: the fundamental interactions between charged polymer chains in confinement. This is a tricky subject partly because at the level of entangled polymer gels, fundamental polymer-polymer interactions do not come up in relevant literature. Past research has instead focused on the collective behavior of these polymer melts, i.e. orientational phase transitions, viscosity and relaxation scaling laws. This is reasonable, since in such dense and entangled systems fundamental interactions are maybe too complex and it is unclear how they relate to macroscopic gel behavior. So, as a first step, I decide to avoid the complexity of a confined polymer gel and instead focus on the interactions (attractive and repulsive) between two polyelectrolyte chains immersed in a good solvent. As a second step, I want to introduce a single wall ( a confining barrier) parallel to the chains and see how the confining potential influences the polymer-polymer interaction. Possibly, a third step would be to include a crowding agent, meant just to occupy space in the solvent, to tip over the repulsion between the polymers into attraction and watch aggregation take place. Below is a FAST draft:

1) Polymer-polymer interactions

In this regime three interactions (at least!) are keeping the chains apart:

Entropic repulsion

Entropic repulsion is inherent to the term 'polymer', since by polymer we mean a large chain of repetitive segments N which perform a thermal, self-avoiding random walk further modified by excluded volume effects and solvent condition. We assume that our polymers are swollen since they are immersed in a good solvent (<math>\chi < 1/2</math>). The end-to-end distance for these polymers will consequently depend exponentially with N, and the value of the exponent is the typical Flory value of 3/5 <math>^1</math> . I will not dwell on the free energy of the single chain, since we are looking for an interaction energy between the two chains.

To get an estimate of that interaction I resort to the prediction for the osmotic pressure between polymer blobs in a semi-dilute polymer regime of concentration <math>\phi</math>. According to our Witten textbook, the osmotic pressure pushing the blobs apart will be<math>^2</math> :

Here A is an experimental parameter depending on the type of polymer and solvent, yet independent on polymer chain length. Thus we obtain an estimate for entropic repulsion in units of <math>k_BT</math>. But am I right to neglect the physics of the single polymer chain and opt for a scaling argument for a semi-dilute polymer solution?

Electrostatic repulsion

To make this discussion relevant to biopolymers (such as cytoskeletal polymers or DNA chains, which invariably posses a surface charge), we have to consider the electrostatic repulsion arising from the surface charge of the two polymer chains. Before we launch into this, it is worth mentioning that charge -much like favorable solvent conditions- further stretches out a polymer chain and partially suppresses thermal fluctuations. This is intuitively understood if we think back to the self avoiding potential (amplified by charge-charge repulsion along chain segments) that stretches out the polymer chain.

To get an estimate of the screened Coulomb interaction between two polyelectrolyte chains immersed in an ionic solvent, we have to solve a Poisson-Boltzmann equation. I don't despair just yet, since a problem of the sort was already cleverly solved by Brenner & Parsegian back in 1974<math>^3</math> . The publication actually addresses the problem of two rigid, cylindrical polyelectrolytes (as opposed to two cylindrical wiggling polyelectrolytes) at mutual angle <math>\theta</math>, but I figure that this approximation is good enough at this point and decide to live with that discrepancy. (I secretly wonder whether it matters a lot and intend to come back to this delicate assumption later in the game). So, Brenner & Parsegian solve the following equation:

This first step is achieved by expanding the exponentials in the first equation and keeping only the leading term of the expansion. <math>\kappa^2</math> is defined as:

<math>\kappa^2 = \frac{8 \pi n e^2}{\epsilon kT}</math>

Here <math>\epsilon</math> is the dielectric constant of the bathing medium and <math>n = \frac{1}{2} \sum n{_i}^0z_i</math> designates the concentration of the ionic species (ions of valence <math>z_i</math> having concentrations <math>n{_i}^0</math>. The potential <math>\Psi</math> varies with the distance r from the rod body as:

<math>\Psi = \frac{2 \nu_h}{\epsilon}K_0(\kappa r)</math>,

where <math>\nu_h</math> is the line-charge density of the rod and <math>K_0</math> is a zeroth order Bessel function. Now this is lightly puzzling since traditionally we assume the electric potential to be a simple exponential charge distribution, but for now I trust the trick. Finally, at angle <math>\theta = 180</math> , the electrostatic repulsion between rods at a distance r from each other is:

Here the constants are dependent on the radius of the cylinders <math>R_A</math>, the dielectric constant of the medium <math>\epsilon</math>, the rod surface charge <math>\sigma</math> and the solvent ionic concentration n. Again, after ploughing through the constants we are left with an energy estimate in terms of <math>k_BT</math>. This is encouraging.

Van der Waals attraction

Apart from the above forces, polarizability effects should be taken into consideration when studying rods in a solvent. The van der Waals potential energy between two rods of identical radius <math>R_A</math> at a distance r from each other is<math>^4</math> :

<math>U_{VdW} = -\frac{AC}{\sqrt{r^3}}</math>

Here C is a constant depending solely on geometric characteristics of the rods, while A is the infamous Hamaker constant which is dependent of optical properties of both rods and of the medium. The Hamaker constant in experimentally determined for many systems in units of <math>k_BT</math>, so we're still in business.

2) Polymer-polymer interactions in the presence of confining potential

Consider a repulsive wall parallel to the polymer system (and NOT vertical as depicted on the image). Then de Gennes<math>^6</math> makes a simple point: the osmotic pressure of the solution <math>\Pi</math>, will again be:

<math>\Pi = k_B T (3.2 A)^{-3} \Phi_1</math>

However, this time <math>\Phi_1</math> is not the concentration of the whole solution but rather a concentration profile. At large distances from the wall, <math>\Phi_1 = \Phi_0</math>, the unconfined concentration of the semi-dilute solution. The first polymer layer, next to the wall, will have a significantly decreased concentration obeying a power law with respect to the ratio <math>\frac{z}{\xi}</math>, where <math>\xi</math> is the mesh size of the network and z the distance from the wall. In other words:

My intuitive understanding is that a repulsive external potential acting on both polymers will attenuate the inter-polymer repulsive forces, and this is what this simple scaling argument also predicts. If the polymers are at close enough distance r, the confining potential might even drive polymer self-assembly? This is intriguing... Can't wait to see how these predictions will measure up to the cold hard math facts!

3) Polymer-polymer interactions in the presence of confining potential and crowding agent

Why include a crowding agent at this point? Because it is certain to do the trick, especially if confinement hasn't! A crowding agent will cause a depletion attraction (essentially an excluded volume effect) that will certainly push the two polymers together. The depletion potential will be dependent on the concentration of the crowding agent in solution as well as the aspect ratio of agent vs. polymer (constant C). The attractive potential will be of the form<math>^5</math> :

<math>V_{depl} = - k_BT \rho_{agent} C</math>

This attraction is certain to overrun all previously mentioned interactions, for high enough <math>\rho</math>.

I hope to be able to find the interaction potentials I'm still missing. Then, to test the waters, I'll choose a polymer system and try to plug in all the numbers! This sounds daunting but if I'm able to pull it off it will be well worth it: it will point at which interactions are relevant and how these balances tip off as we change various parameters (like polymer-polymer distance or polymer-wall distance). Amen!